Systems and methods for adaptive multiresolution signal analysis with compact cupolets

ABSTRACT

Systems and methods for signal analysis using orbits of a chaotic system are provided. For example, a multiresolution analysis may be constructed and cupolets may be used to approximate arbitrary signals and compress images. Cupolets may be phase transformed to produce compact cupolets that are well-suited for producing sharp changes in signals, or to produce compact cupolets that are more oscillatory and have less or no sharp global maximum amplitudes. Alternatively, cupolets may be phase transformed to allow for optimal or near optimal adjustment to fit a signal.

CROSS-REFERENCE TO OTHER PATENT APPLICATIONS

This application claims the benefit of U.S. Provisional PatentApplication No. 60/874,582, filed Dec. 12, 2006, and U.S. ProvisionalPatent Application No. 60/899,722, filed Feb. 5, 2007, each of which ishereby incorporated by reference herein in its entirety.

FIELD OF THE INVENTION

This can relate to systems and methods for signal analysis using orbitsof a chaotic system and, more particularly, to systems and methods foradaptive multiresolution signal analysis using cupolets.

BACKGROUND OF THE DISCLOSURE

Periodic orbits of a chaotic system are a rich source of qualitativeinformation about the dynamical system. For example, the presence ofunstable periodic orbits on the attractor is a typical characteristic ofa chaotic system. The abundance of unstable periodic orbits have beenutilized in a wide variety of theoretical and practical applications foranalyzing, compressing, and processing signals. Therefore, it isdesirable to be able to provide new and improved systems and methods foranalyzing signals using orbits of chaotic systems.

SUMMARY OF THE DISCLOSURE

Systems and methods for signal analysis using orbits of a chaotic systemare provided.

According to one embodiment, a method for operating on a cupolet toproduce bases for representing a signal includes phase transforming thecupolet into a compact cupolet and extending the compact cupoletnon-periodically over the entire real line by defining the compactcupolet to be zero outside the window of the compact cupolet. The methodalso includes applying a transform to the extended compact cupolet toobtain at least a first basis, using the first basis for a firstresolution-level, and selecting at least one basis element from thefirst basis to approximate the signal at a first resolution.

According to another embodiment, a method for operating on a cupolet toproduce bases for representing a signal includes phase transforming thecupolet into a compact cupolet and down-sampling the compact cupolet bya factor of the length of the window of the compact cupolet. The methodalso includes padding the down-sampled compact cupolet to have the samelength as the length of the window of the compact cupolet, extending thepadded cupolet periodically to the entire real line, sampling theextended cupolet to obtain at least a first basis, using the first basisfor a first resolution-level, and selecting at least one basis elementfrom the first basis to approximate the signal at a first resolution.

According to yet another embodiment, a method for operating on a cupoletto produce bases for representing a signal includes phase transformingthe cupolet into a compact cupolet and down-sampling the compact cupoletby a first factor of the length of the window of the compact cupolet toprovide a first down-sampled cupolet. The method also includes paddingthe first down-sampled cupolet to fit in the window of the compactcupolet, shifting the padded first down-sampled cupolet by the firstfactor to generate at least a first basis, using the first basis for afirst resolution-level, and selecting at least one basis element fromthe first basis to approximate the signal at a first resolution.

According to still yet another embodiment, a method of converting acupolet from an oscillatory state into a compact state includes phasetransforming the cupolet into a compact cupolet by aligning at least aportion of the frequency components of the cupolet around a singleglobal maximum.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other advantages of the invention will be apparent uponconsideration of the following detailed description, taken inconjunction with accompanying drawings, in which like referencecharacters refer to like parts throughout, and in which:

FIG. 1 is a flowchart of an illustrative process for generatingcontrolled stabilized periodic orbits constructed in accordance with theprinciples of the invention;

FIG. 2 is a plot of a double scroll oscillator generated from systemsand methods constructed in accordance with the principles of theinvention;

FIG. 3 is a top view of the intersection of a Poincare surface with aportion of the double scroll oscillator of FIG. 2 generated from systemsand methods constructed in accordance with the principles of theinvention;

FIG. 4 is a plot of a coding function r(x) around the double scrolloscillator of FIG. 2 generated from systems and methods constructed inaccordance with the principles of the invention;

FIG. 5 is a plot of a periodic orbit of the double scroll oscillator ofFIG. 2 generated from systems and methods constructed in accordance withthe principles of the invention;

FIG. 6 shows illustrations of cupolets and corresponding compactcupolets generated from systems and methods constructed in accordancewith the principles of the invention;

FIG. 7 shows illustrations of basis elements of a space generated fromsystems and methods constructed in accordance with the principles of theinvention;

FIG. 8 shows illustrations of cupolets approximating arbitrary signalsusing various resolution-levels generated from systems and methodsconstructed in accordance with the principles of the invention;

FIG. 9 shows illustrations of an image compressed using variousresolution-levels generated from systems and methods constructed inaccordance with the principles of the invention;

FIG. 10 shows illustrations of compact cupolets at variousresolution-levels generated from systems and methods constructed inaccordance with an embodiment of the invention;

FIG. 11 shows illustrations of approximations of arbitrary signals usingvarious resolution-levels and corresponding spectral values generatedfrom systems and methods constructed in accordance with the principlesof the invention;

FIG. 12 shows illustrations of approximations of an arbitrary signalusing various numbers of basis elements generated from systems andmethods constructed in accordance with the principles of the invention;

FIG. 13 shows illustrations of an image approximated using variousnumbers of basis elements generated from systems and methods constructedin accordance with the principles of the invention;

FIG. 14 shows illustrations of approximations of an arbitrary signalusing various resolution-levels generated from systems and methodsconstructed in accordance with the principles of the invention; and

FIG. 15 shows illustrations of an image approximated using variousresolution-levels generated from systems and methods constructed inaccordance with the principles of the invention.

DETAILED DESCRIPTION OF THE DISCLOSURE

Control schemes for stabilizing unstable periodic orbits of a chaoticsystem are provided herein. The techniques allow for the creation ofmany periodic orbits. These approximate chaotic unstable periodic orbitsmay be referred to herein as Chaotic Unstable Periodic Orbits(“cupolets”). These orbits can be passed through a phase transformationto a compact cupolet state that can possess a wavelet-like structure andcan be used to construct adaptive bases. This cupolet transformation canbe an alternative to Fourier and wavelet transformations. The cupolettransformation can also provide a continuum between Fourier and wavelettransformations, and can be used in a variety of applications, such asdata and music compression, as well as image and video processing.

An advantage of using cupolets over traditional Fourier and wavelettransforms for creating adaptive bases in accordance with the inventionis that cupolets can be used to generate a wide variety of differentwave forms, ranging from a simple sine-like wave with only one dominantspectral line to very complex wave forms with lots of harmonics, forexample. These cupolets can be produced with very little information(e.g., using only 16 control bits, although any suitable number ofcontrol bits may be used), and mathematically, any complete basis can beused. However, efficiency of a chosen basis for compression purposes maybe tied to convergence rates. Real world signals, such as signalsobtained from musical instruments and images with edges anddiscontinuities, may have very complicated spectra. Therefore, using amore complex signal can help to achieve a faster convergence rate.

One chaotic control approach of the invention may be adapted from acommunication scheme developed by Hayes, Grebogi, and Ott, as describedin more detail below, which uses small perturbations applied on acontrol surface to steer trajectories of a double scroll oscillatoraround each of two loops in the attractor, which may be labeled 0 and 1to correspond to bit values. Therefore, an analog signal may be obtainedand the bit value can be determined by observing whether the oscillationis above or below a reference value. “Parker, A. T.: ‘Topics In ChaoticSecure Communication,’ PhD. thesis, University of New Hampshire (1999),”“Short, K. M., Parker, A. T.: ‘Security Issues In ChaoticCommunications,’ Paper presented at the SIAM Conf. on Dynamical Systems,Snowbird, Utah, May 23-27 (1999),” and “U.S. Pat. No. 7,110,547 entitled“Method and Apparatus for the Compression and Decompression of ImageFiles Using a Chaotic System,” each of which is hereby incorporated byreference herein in its entirety, shows that this scheme can be adaptedfor secure communication. A receiver can be initialized remotely bysending sequences of initializing codes that can cause a chaotic systemto stabilize onto a cupolet, regardless of the initial state of thesystem.

However, all the different dynamical behaviors of the system may beeasily accessible via small controls. The techniques of the inventionmay be used to produce cupolets that closely approximate periodic orbitsof the chaotic system. The orbits may be produced with smallperturbations, which in turn may suggest that these orbits are shadowingtrue periodic orbits.

These cupolets can be utilized in constructing an adaptivemultiresolution analysis of signals. Various approaches may be used toapproximate arbitrary signals at different resolution-levels accordingto the invention. Adaptive image compression may provide the ability toaccess an image at different resolutions, ranging from a low qualityimage with a small size to a high quality image with a bigger size, forexample. The multiresolution analysis may allow for the lowestresolution levels to capture the coarse structure of an image while thefiner detail may be carried in the higher resolution levels. This can beuseful, for example, in web-based applications. In a progressivedownload mode, for example, an image can arrive earlier and can bequickly displayed in lower quality and, if the network bandwidth allows,the finer resolution layers can be added as they arrive to provide thefine details. Similarly, smaller images, such as thumb-nails, can beaccomplished with coarse resolutions, as their small size may limit therequired resolution.

The control schemes may be used to stabilize unstable periodic orbits ofa chaotic system, such as a double scroll system. Multiresolutionanalysis may be constructed such that cupolets of the chaotic system canbe used to approximate arbitrary signals and compress images.

In view of the foregoing, systems and methods for signal analysis usingorbits of a chaotic system are provided and described with reference toFIGS. 1-15.

FIG. 1 shows an illustrative process 100 for generating controlledstabilized periodic orbits of a chaotic system in accordance with theinvention. Control signals can be utilized in a chaotic system to inducethe chaotic system to settle onto periodic orbits that would otherwisebe unstable (e.g., aperiodic). A control signal may be relatively smallin length (e.g., approximately 16 bits), but the resultant periodicwaveforms can include more than 200 harmonics in their spectrum. Thedifference in size between the control signals and the resultantwaveforms may be utilized to create a compression scheme. For example,the compression achieved may be related to the ratio of the number ofinformation bits required to define the spectral lines of the waveformand the number of bits required to define the cupolet.

Waveforms produced by the chaotic signal generator may be, for example,cupolets. Cupolets may naturally carry a structure present in varioustypes of signals, such as speech, music, and image signals. Accordingly,cupolets can be used individually, or combined with one another, tomodel such signals.

Process 100 of FIG. 1 may begin by choosing a chaotic system at step102. One type of chaotic system is the double-scroll oscillator, whichmay be defined by the following set of nonlinear differential equationsthat form a 3-variable system, for example, as described by “S. Hayes,C. Grebogi, and E. Ott: ‘Communicating with Chaos,’ Phys. Rev. Lett. 70,3031 (1993),” which is hereby incorporated by reference herein in itsentirety:

${C_{1}\frac{\mathbb{d}V_{C\; 1}}{\mathbb{d}t}} = {{G\left( {V_{C\; 2} - V_{C\; 1}} \right)} - {g\left( V_{C\; 1} \right)}}$${C_{2}\frac{\mathbb{d}V_{C\; 2}}{\mathbb{d}t}} = {{G\left( {V_{C\; 1} - V_{C\; 2}} \right)} + i_{L}}$${L\frac{\mathbb{d}i_{L}}{\mathbb{d}t}} = {- V_{C\; 2}}$ where${g(V)} = \left\{ \begin{matrix}{{m_{1}V},} & {{- B_{p}} \leq V \leq B_{p}} \\{{{m_{0}\left( {V + B_{p}} \right)} - {m_{1}B_{p}}},} & {V \leq {- B_{p}}} \\{{{m_{0}\left( {V - B_{p}} \right)} + {m_{1}B_{p}}},} & {V \geq B_{p}}\end{matrix} \right.$

Here, g(v) represents a nonlinear negative resistance component, and C₁,C₂, L, G, m₀, m₁, and B_(p) are constant parameters. In someembodiments, these equations can be used to build an analog circuit or adigital circuit. In other embodiments, these equations can be simulatedon a computer as software. For example, a programmable logic device maybe utilized to embody the equations in hardware. If a circuit is built,the variables V_(C1) and V_(C2) may be voltages, and i_(L) may be acurrent. In the equations, the variables may be real and continuous,while the output of a software simulation may produce a sampledwaveform.

A chaotic system, such as, for example, a double-scroll oscillator, maysettle down to, and may be bounded by, an attractor. The system mayregularly settle down to the same attractor no matter what initialconditions were used to set the system. In the 3-variable systemprovided by the above equations, these attractors may be ribbon-likestructures that stretch and fold upon themselves and remain confined toa box. The actual state of the 3-variable system may be determined bythe instantaneous value of the system variables (e.g., V_(C1), V_(C2),and i_(L)). The state of the system may be defined by the joint state ofthe variables (i.e., the state may be a 3-vector) and, therefore, one ofthe variables may have the same value more than once. But, whenever thisoccurs, at least one of the other variables may have a different orunique value, such that if we plot the state of the system as a point inthree-dimensional space, then no point may be duplicated in someembodiments. Therefore, the values of these variables may never repeatsuch that an aperiodic system may be provided, for example.

While the chaotic attractors may be aperiodic structures, the attractorscan have an infinite number of unstable periodic orbits embedded withinthem. Control signals may be provided to stabilize these orbits byperturbing the state of the system in certain fixed locations by aparticular amount. Using the above equations as an example, theattractor that results from a numerical simulation using the parametersC₁=1/9, C₂=1, L=1/7, G=0.7, m₀=−0.5, m₁=−0.8, and B_(p)=1 may have twolobes, each of which may surround an unstable fixed point, as shown inFIG. 2.

Because of the chaotic nature of the oscillator's dynamics, it ispossible to take advantage of sensitive dependence on initial conditionsby carefully choosing small perturbations to direct trajectories aroundeach of the lobes of the attractor. In this way, steering thetrajectories around the appropriate lobes of the attractor, suitablylabeled 0 and 1 in FIG. 2, for example, can generate a desired bitstream. This makes it possible, through the use of an initializationcode, to drive the chaotic system onto a periodic orbit that is used toproduce a basic waveform. It should be noted that other embodimentscould have more than two lobes, in which each lobe is labeled 0 or 1 ora symbol from any chosen symbol set.

There are a number of ways to control the chaotic oscillator to moreprecisely specify the bits 0 and 1. In one embodiment, a Poincaresurface of section may be defined on each lobe by intersecting theattractor with the half planes

i_(L) = ±GF, υ_(C₁) ≤ F, where$F = {\frac{B_{p}\left( {m_{0} - m_{1}} \right)}{\left( {G + m_{0}} \right)}.}$The lobes and control planes may be assigned values 0 and 1, as shown inFIG. 2, such that a sequence of binary digits may be recorded when atrajectory passes through a control plane.

These half planes may intersect the attractor with an edge at theunstable point at the center of each lobe, for example. The Poincaresurface may be two-dimensional, but because the attractor is also nearlytwo-dimensional close to this surface, the intersection between theattractor and the Poincare surface may be approximately one-dimensional.For example, FIG. 3 shows a top view of the intersection of the Poincaresurface with one of the lobes of the double scroll oscillator of FIG. 1.The asterisks of FIG. 3 mark the intersection of the trajectory with thePoincare half plane. In one embodiment, this set of points may beapproximated by a line extending from the unstable fixed point fittedwith the least squares method, for example.

In one embodiment, this line segment may then be partitioned into Pequally spaced bins and the distance x from the center of each bin tothe center of the corresponding lobe may be recorded. Each one of thesepoints may then be used as a starting point and computer simulations maybe run without control. The obtained trajectory may result in a symbolicsequence that is the sequence of lobes visited by the trajectory. Thesesymbolic sequences may then be stored in a bit register. The symbolicstate of the system can be represented by a function r which maps thesymbolic state space coordinate x on the Poincare surface to a binaryrepresentation of the symbol sequence obtained from x.

For example, the function r(x) may be defined that takes any point oneither section and returns the future symbolic sequence for trajectoriespassing through that point. If b₁, b₂, b₃, . . . represent the lobesthat are visited on the attractor (so b₁ is either a 0 or a 1), and thefuture evolution of a given point x₀ is such that x₀→b₁, b₂, b₃, . . . ,b_(N) for some number N of loops around the attractor, for example, thenthe function r(x) may be chosen to map x₀ to an associated binaryfraction, so r(x₀)=0. b₁b₂b₃ . . . b_(N), where this represents a binarydecimal (base 2). Then, when r(x) is calculated for every point on thecross-section, the future evolution of any point on the cross-sectionmay be known for N iterations. The resulting function is shown in FIG.4, where r(x) has been calculated for 12 loops around the attractor.This function r(x) may be referred to as “the coding function,” and maybe defined by

${r(x)} = {\sum\limits_{n = 1}^{\infty}{\frac{b_{n}}{2^{n}}.}}$

The coding function r(x) may have to be truncated to some finite valueas it cannot be tracked for all future time. The truncated version ofr(x) may be denoted by the following equation r_(N)(x), where theinfinite sum is truncated at n=N.

${r_{N}(x)} = {\sum\limits_{n = 1}^{N}{\frac{b_{n}}{2^{n}}.}}$

In order to control the system to follow a desired symbolic sequence,the scheme may require running the simulation and waiting until thetrajectory crosses a Poincare surface, for example at x₀. The value ofr(x₀) may yield the future symbolic sequence followed by the currenttrajectory for N loops. If a different symbol in the Nth position of thesymbolic sequence is desired, r(x) can be searched for the nearest pointon the section that will produce the desired symbolic sequence. Thetrajectory can be perturbed to this new point, and it may continue toits next encounter with a surface. This procedure can be repeated asmany times as is desirable.

In some embodiments, the calculation of r(x) may be done discretely bydividing up each of the cross-sections into a certain amount ofpartitions or bins and by calculating the future evolution of thecentral point in the partition for up to a certain number of loopsaround the lobes. As an example, controls may be applied so that effectsof a perturbation to a trajectory can be evident after only a certainamount of loops around the attractor (e.g., after only 5 loops). Inaddition to recording r(x), a matrix M may be constructed that containsthe coordinates for the central points in the bins, as well asinstructions concerning the controls at these points. These instructionsmay simply describe how far to perturb the system when it is necessaryto apply a control. For example, at an intersection of the trajectorywith a cross-section, if r(x₀) indicates that the trajectory will traceout the sequence 10001, and sequence 10000 is desired, then a search maybe made for the nearest bin to x₀ that will give this sequence, and thisinformation may be placed in matrix M. If the nearest bin is not unique,then there may be an agreement about which bin to take (e.g., the binfarthest from the center of the loop).

Because the new starting point after a perturbation may have a futureevolution sequence that may differ from the sequence followed by x₀ byat most the last bit, only two options may need to be considered at eachintersection: control or no control. In an analog hardwareimplementation, the perturbations may be applied using voltage changesor current surges. In a software implementation, the control matrix Mmay be stored along with the software computing the chaotic dynamics sothat when a control perturbation is required, the information may beread from matrix M.

Microcontrols may also be used. For example, in a software embodiment,each time a trajectory of the chaotic system passes through across-section, the simulation may be backed-up one time step, and theroles of time and space may be reversed in a Runge-Kutta solver, forexample, so that the trajectory can be integrated onto the cross-sectionwithout any interpolation. Then, at each intersection where no controlis applied, the trajectory may be reset so that it starts at the centralpoint of whatever bin it is in. This resetting process can be consideredthe imposition of microcontrols. It may remove any accumulation ofround-off error and may minimize the effects of sensitive dependence oninitial conditions. It also may have the effect of restricting thedynamics of the chaotic attractor to a finite subset of the full chaoticattractor, although the dynamics may still visit the full phase space.These restrictions can be relaxed by calculating r(x) and matrix M togreater precision at the outset.

As also shown in process 100 of FIG. 1, the next step 104 may be theimposition of an initialization code on the chaotic system. Theinitialization code may drive the chaotic system onto a periodic orbitand may stabilize the otherwise unstable periodic orbit. Morespecifically, the chaotic system may be driven onto a periodic orbit bysending it a repeating code. Different repeating codes may lead todifferent periodic orbits. For a large class of repeating codes, theperiodic orbit reached may be dependent only on the code segment that isrepeated, and not on the initial state of the chaotic system (althoughthe duration of time to get on the periodic orbit can vary depending onthe initial state). Consequently, it is possible to send aninitialization code that drives the chaotic system onto a known periodicorbit.

These special repeating codes may lead to unique periodic orbits for allinitial states, so that there may be a one-to-one association between arepeating code and a periodic orbit. However, for some repeating codes,the periodic orbits themselves may change as the initial state of thechaotic system changes. Consequently, repeating codes can be dividedinto two classes, initializing codes and non-initializing codes. Thelength of each periodic orbit may be an integer multiple of the lengthof the repeating code, since periodicity may be attained only when boththe current position on the cross-section as well as the currentposition in the repeating code may be the same as at some previous time.To guarantee that the chaotic system is on the desired periodic orbit,it may be sufficient that the period of the orbit be the length of thesmallest repeated segment of the initializing code.

As is further shown in process 100 of FIG. 1, the next step 106 may begenerating a basic waveform from the periodic orbit. In one embodiment,this may be a one-dimensional, periodic waveform, for each periodicorbit, by taking the x-, y-, or z-component (or any combination of them)of the periodic orbit over time. By sampling the amplitude of thewaveform over time (e.g., using audio standard PCM 16), a digitalversion can be produced. These basic waveforms can be highly complex andcan have strong harmonic structure. The basic waveforms can have morethan 50 strong harmonics for some initialization codes, and an importantfactor that contributes to the performance of the signal analysis (e.g.,signal compression) may be that complex basic waveforms with many strongharmonics can be produced with the same number of bits in theinitialization code as simpler basic waveforms with only a fewharmonics. This is indicative of the potential for compression inherentin this system since complex basic waveforms may be produced as easilyas simple basic waveforms. This may only be possible because of thenonlinear chaotic nature of the dynamical system.

The chaotic system can be implemented entirely in software. The chaoticsystem in such an implementation may be defined by a set of differentialequations governing the chaotic dynamics (e.g., the double scrollequations described above). The software may utilize an algorithm tosimulate the evolution of the differential equations (e.g., the fourthorder Runge-Kutta algorithm).

The chaotic system can also be implemented in hardware. The chaoticsystem may still be defined by a set of differential equations, butthese equations may then be used to develop an electrical circuit thatcan generate the same chaotic dynamics. The procedure for conversion ofa differential equation into an equivalent circuit is well-known and canbe accomplished with analog electronics, microcontrollers, embeddedcentral processing units (“CPUs”), digital signal processing (“DSP”)chips, or field programmable gate arrays (“FPGAs”), as well as otherdevices known to one skilled in the art, configured with the properfeedbacks. The control information may be stored in a memory device, andcontrols may be applied by increasing voltage or inducing small currentsurges in the circuit, for example.

Therefore, the process of perturbing or not perturbing the trajectorycan be described by a binary sequence, with 1 meaning apply the controland 0 meaning no control, for example. If a control is repeatedlyapplied to the system, after an initial transient state, the trajectorymay close up on itself to form a unique periodic orbit.

When the binary sequence of controls and microcontrols are applied inthis manner, there may only be a finite number of orbits on theattractor, so the periodicity of the dynamics may eventually beguaranteed even if no controls are applied. However, when theinitializing controls are applied, the system may stabilize onto theperiodic orbits, sometimes referred to as cupolets. For example, acompressed initializing code (e.g., 01011) may be repeated for thedouble-scroll oscillator of FIG. 2. The chaotic dynamics of the systemmay be driven onto the periodic orbit shown in FIG. 5. This periodicorbit may be stabilized by the control code, and may be a period 5cupolet, for example.

A number of the control strings may cause the chaotic system tostabilize onto a periodic orbit, and these periodic orbits may be inone-to-one correspondence with the control string used (and may beindependent of the initial state of the system). Furthermore, for agiven gate size for a control scheme and for a fixed number N of controlbits, a finite, but sufficiently large number of cupolets can begenerated.

Once a cupolet is stabilized, for example, the cupolet may form a closedloop that tracks around the attractor and may be defined by the statevariables (e.g., V_(C1), V_(C2), and i_(L)), for example. The conversionto a one-dimensional waveform can be done in a circuit implementation bytaking the output of one of the voltage or current measurements, forexample. If performed in software, a digitized waveform can be produced,for example, by sampling one of the state variables. The term cupoletcan be used to, for example, represent both the periodic orbit in threedimensions and the one-dimensional waveforms that it may produce.

Alternatively or additionally to double scroll oscillator systems, otherchaotic systems, such as Rossler systems, Lorenz systems, and generalunimodal maps, may also be used to generate cupolets according to theinvention. For example, cupolets generated using a Rossler system mayhave less interesting harmonics, but may be easier to process.

Cupolets may be useful in a variety of applications, and particularly incompression of speech, music, images and video, for example, asdescribed in “Short, K. M., Garcia, R. A., Daniels, M., Curley, J.,Glover, M.: ‘An Introduction to the Koz Scalable Audio CompressionTechnology,’ AES 118th Convention preprint: 6446 (May 2005)” and “Short,K. M., Garcia, R. A., Daniels, M.: ‘Scalability in Koz Audio CompressionTechnology,’ AES 119th Convention preprint: 6598 (October 2005),” eachof which is hereby incorporated herein in its entirety. However, theproperties of cupolets can also be utilized to approximate discretesignals. Therefore, according to the invention, a multiresolutionanalysis may be constructed and cupolets may be used to approximatearbitrary signals and compress images, for example.

Two of the most effective signal analysis techniques for compression arewindowed Fourier and wavelet transforms. The building blocks of windowedFourier analysis are sines and cosines or their complex equivalents.When applied to signal processing of digital data, the windowed Fourieranalysis may be done with sliding data windows. On the other hand,wavelet analysis is analysis based on scale. In some ways, waveletanalysis can be an alternative to classical Fourier analysis. In waveletanalysis, the building block can be referred to as a mother wavelet andmay generally be compact and oscillatory. There are three basic waveletoperators that can play the role of sliding windows in Fourier analysis.These three operators are: (1) translation by h, defined by (τ_(h)f)(x)=f(x−h); (2) dilation by r>0, defined by (ρ_(r)f) (x)=f(rx); and (3)modulation by m, defined by (μ_(m)f)(x)=e^(imx)f(x). These operators maybe applied to a mother wavelet to produce other wavelets.

Cupolets can be used in a similar manner to produce a multiresolutionanalysis according to the invention. Cupolets can be transformed betweenan oscillatory state similar to the sinusoidal basis of the Fourieranalysis and a “compact cupolet” state that is wavelet-like. Althoughdescribed in more detail below, a compact cupolet state may be referredto herein as a state where the energy of a cupolet has been concentratedaround a single global maximum.

The space of real-valued N-periodic functions of a discrete variable maybe denoted by

^(N), as shown by the following:

^(N) ={f:

→

|f(n+N)=f(n),nεN

},where N may represent the number of discrete samples in one period ofthe function.

If γ(t)ε(

³)^(N) is a periodic orbit of a chaotic system, such as a periodic orbitof the chaotic system described above with respect to FIGS. 1-5 (i.e., acupolet of the double scroll oscillator), and assuming that γ(t)=(x(t),y(t), z(t)), the discrete Fourier transform of one of the components ofγ may be computed (e.g., discrete Fourier transform {circumflex over(x)} may be computed for component x). α_(k) may represent the Fouriercoefficients of the signal x, as shown by the following:

$\alpha_{k} = {{\frac{1}{2\pi}{\hat{x}\left( {\frac{2\pi}{N}k} \right)}} = {\frac{1}{N}{\sum\limits_{n = 0}^{N - 1}{{x(n)}{{\exp\left( {{- {\mathbb{i}}}\;{k\left( \frac{2\pi}{N} \right)}n} \right)}.}}}}}$For a real valued signal xε

^(N), the Fourier coefficients α_(k), k=0, . . . , N−1, are complexvalued numbers, as shown by the following:α_(k)=|α_(k)|exp(iφ(k)),where φ(k) in the above equation is the phase term.

To convert the cupolet into a compact state according to the invention,a phase transformation can be performed on the cupolet. This may alignall of the frequency components so that they can add constructively at acertain position of the window (e.g., the center of the window). Inorder to concentrate the energy of the cupolet at the p-th position ofthe window, the phase term can be changed to:

$2\pi\; k\;{\frac{p}{N}.}$For the case where

${P = \frac{N}{2}},$the final phase term may be of the form πk. Furthermore, α _(k) may bedefined as:{tilde over (α)}_(k)=|α_(k)|exp(iπk),and the signal {tilde over (x)}ε

^(N) may be defined by:

${{\overset{\sim}{x}(n)} = {{\sum\limits_{k = 1}^{N}{{\overset{\sim}{\alpha}}_{k}{\exp\left( {{\mathbb{i}}\;{k\left( \frac{2\pi}{N} \right)}n} \right)}\mspace{14mu}{for}\mspace{14mu} n}} = 0}},\ldots\mspace{14mu},{N - 1.}$In this way, a discrete periodic signal {tilde over (x)} with a period Nmay be obtained that has most of its energy concentrated in the centerof the window, for example.

A few examples of a cupolet and its corresponding compact form are shownin FIG. 6, for example. The three examples of compact cupolets shown inthe graphs of the second column of FIG. 6 (i.e., parts (d), (e), and(f)) may be created by performing this phase transformation of theinvention on the respective cupolets shown in the graphs of the firstcolumn of FIG. 6 (i.e., parts (a), (b), and (c)).

These compact cupolets may be wavelet-like signals. Various methods ofoperating on a compact cupolet to produce bases for functionrepresentation are provided according to the invention. The optimumselection of the cupolet may be accomplished in many suitable ways. Inone embodiment, the harmonic structure of the signal may be examined andthen the cupolet that most closely matches the spectrum may be chosen asthe optimum cupolet. For example, the cupolet with a power spectrumclosest to the spectrum of the signal to be analyzed in a least squaressense, or any other type of minimization function, for example, may bechosen.

According to one embodiment of the invention, operating on a compactcupolet to produce bases for function representation may be accomplishedby constructing a basis using a recursive process similar to the processof creating a Walsh basis. A discussion on Walsh transforms may befound, for example, in “Beer, T.: ‘Walsh Transforms,’ American Journalof Physics 49 (1981) 466-472,” which is hereby incorporated by referenceherein in its entirety. According to another embodiment of theinvention, operating on a compact cupolet to produce bases for functionrepresentation may be accomplished by defining and conducting a partialperiodic multiresolution analysis and studying the properties of thebasis elements at different resolution-levels. According to yet anotherembodiment of the invention, operating on a compact cupolet to producebases for function representation may be accomplished by conducting aperiodic multiresolution analysis that may be constructed with eachresolution-level created by translations of an appropriately chosenscaling function.

In each case, arbitrary signals may be approximated at differentresolution-levels and a sample image may be compressed. The image may bea standard image widely used in the image compression community, forexample. Any signal that can be sampled may be approximated according tothe invention, including, but not limited to, signals found in media(e.g., voice, music, images, and video), measurement outputs of ameasurement in a system (e.g., voltage measurements from some point in acircuit or average monthly sunspot activity over a period of time), andcombinations thereof. Whenever a basis is determined, any signal in thespace characterized by the basis may be represented. Some bases mayprovide a rapidly converging representation, such that therepresentation may be truncated without losing too much valuableinformation. Such a truncated representation may provide a compressedrepresentation.

Although each of these approaches may produce acceptable results, evenbetter quality may be achieved by manipulating the phase of the cupolet.A complexification process is provided according to the invention thatcan create a degree of phase flexibility and can help to avoid theartifacts that may appear in the image due to compression and theinherent grid alignment that may occur in other approaches. This processof complexification may force the cupolet to move into a potentiallynon-compact state and may adjust the cupolet to match the signal in thebest possible way with one (complex) degree of freedom.

It is to be noted that the basis generation techniques of the inventionmay be known by both the transmitter and the receiver of the compressedimage, through storage in ROM or by regeneration in software, forexample. Compression may occur at a functional level when an N-pointsampled signal can be regenerated to a desired quality level with fewerthan N basis elements. It is also to be noted that compression, asreferred to herein, may generally be understood to be function-levelcompression. Function-level compression is still one step removed fromthe bit-level compression that should be examined to determine thegeneral and most efficient compression method. In practicalapplications, bit-level compression may factor in the number of bitsused to represent a basis element. The various techniques provided bythe invention for operating on a compact cupolet to produce bases forfunction representation are presented without choosing the mostefficient compression method, as a method that may appear less efficientat a functional level, may allow a greater bit-wise compression due toseveral factors, such as patterns in the coefficients, for example.

According to one embodiment of the invention, operating on a compactcupolet to produce bases for function representation may be accomplishedby constructing a basis using a recursive process similar to the processof creating a Walsh basis. Therefore, a method of creating a Walsh basiscan be used with compact cupolets to approximate elements of the spaceof discrete functions over N=2^(n) samples,

^(N). For example, operating on a compact cupolet of size N, producingbases for function representation may be accomplished as follows.

First, the cupolet may be extended non-periodically over the entire realline by defining the cupolet to be zero outside the window. Next,transforms, such as Walsh transforms, for example, may be applied tothis extension of the cupolet to obtain a chain of linear spaces. Eachof the linear spaces can be regarded as a resolution-level, and eachresolution-level may be used to approximate signals. Finally, in orderto get better approximations, the cupolets may be sent through anonlinear phase deformation, which may adjust the cupolets in a leastsquares sense, for example, to approximate the signal. This process maybe referred to as a complexification process.

In one example, let ψ be a given compact cupolet defined over a samplewindow, assume N=2^(n) samples are taken in the window, and also let ψ₀be defined as zero outside the window. A sequence of functions ψ_(j),j=0, 1, . . . , n−1, may be defined as follows:

${\psi_{{2j} + 2}(x)} = {{\psi_{j}\left( {2x} \right)} + {\left( {- 1} \right)^{j}{\psi_{j}\left\lbrack {2\left( {x - \frac{N}{2}} \right)} \right\rbrack}}}$${\psi_{{2j} + 1}(x)} = {{\psi_{j}\left( {2x} \right)} - {\left( {- 1} \right)^{j}{{\psi_{j}\left\lbrack {2\left( {x - \frac{N}{2}} \right)} \right\rbrack}.}}}$Using this defined sequence of functions, x may be taken to range overthe N samples in the sample window, with the extension of the functionbeing zero outside the window.

In one example, [|Ψ₀|Ψ₁|Ψ₂| . . . |Ψ_(j)], where Ψ_(j), j=1, . . . ,n−1, may be a matrix whose columns may be ψ₂ ^(j−1), . . . , ψ₂^(j)(i.e., Ψ_(j)=[ψ₂ ^(j−1) . . . ψ₂ ^(j)]. One can easily verify thatthe columns of Ψ_(j) may construct an orthogonal set. By letting V^(j)be the closure of the linear span of the columns of the matrix[Ψ₀|Ψ₁|Ψ₂| . . . |Ψ_(j)], in one embodiment, a sequence of linear spacesV⁰⊂V¹⊂ . . . ⊂V^(n−1)=

^(2n) may be obtained. The basis elements for the space Ψ², for example,are shown in FIG. 7. A direct application of this basis can produce afair representation of an image. However, there may be strong andpersistent artifacts where the basis functions overlap. Therefore, adirect use of this adapted Walsh transform may not be optimal. Instead,this technique may be extended.

For example, given an integer 0≦j≦n−2, the basis elements in the spaceV^(j) may be complexified as follows. First, let ψ_(j) be a matrix whosecolumns are the Fourier coefficients of the corresponding column inψ_(j). That is, let{circumflex over (Ψ)}_(j)=[{circumflex over (ψ)}₂ _(j−1) . . .{circumflex over (ψ)}₂ _(j−1) ],where {circumflex over (ψ)}_(k), k=2^(j+1), . . . , 2^(j)−1 are columnvectors that may contain the Fourier coefficients of ψ_(k). Then,{circumflex over (Ψ)}_(j) may be written as{circumflex over (Ψ)}_(j)={circumflex over (Ψ)}_(j) ⁺+{circumflex over(Ψ)}_(j) ⁻,where {circumflex over (Ψ)}_(j) ⁺ and {circumflex over (Ψ)}_(j) ⁻ may betwo matrices whose columns consist of positive frequencies and negativefrequencies of columns of {circumflex over (Ψ)}_(j). For example, if{circumflex over (Ψ)}_(j) is represented by the matrix

${{\hat{\Psi}}_{j} = \begin{bmatrix}a_{00} & a_{10} & \ldots & a_{{({2^{j} - 1})}0} \\a_{01} & a_{11} & \ldots & a_{{({2^{j} - 1})}1} \\\vdots & \vdots & \; & \vdots \\{\overset{\_}{a}}_{02} & {\overset{\_}{a}}_{12} & \ldots & {\overset{\_}{a}}_{{({2^{j} - 1})}2} \\{\overset{\_}{a}}_{01} & {\overset{\_}{a}}_{11} & \ldots & {\overset{\_}{a}}_{{({2^{j} - 1})}1}\end{bmatrix}},$then {circumflex over (Ψ)}_(j) ⁺ and {circumflex over (Ψ)}_(j) ⁻ may berepresented by the matrices

${\hat{\Psi}}_{j}^{+} = {\begin{bmatrix}a_{00} & a_{10} & \ldots & a_{{({2^{j} - 1})}0} \\a_{01} & a_{11} & \ldots & a_{{({2^{j} - 1})}1} \\\vdots & \vdots & \; & \vdots \\0 & 0 & \ldots & 0 \\0 & 0 & \ldots & 0\end{bmatrix}\mspace{14mu}{and}}$${\hat{\Psi}}_{j}^{-} = {\begin{bmatrix}0 & 0 & \ldots & 0 \\0 & 0 & \ldots & 0 \\\vdots & \vdots & \; & \vdots \\{\overset{\_}{a}}_{02} & {\overset{\_}{a}}_{12} & \ldots & {\overset{\_}{a}}_{{({2^{j} - 1})}2} \\{\overset{\_}{a}}_{01} & {\overset{\_}{a}}_{11} & \ldots & {\overset{\_}{a}}_{{({2^{j} - 1})}1}\end{bmatrix}.}$Next, the inverse Fourier transform of the columns of the matrices{circumflex over (Ψ)}_(j) ⁺ and {circumflex over (Ψ)}_(j) ⁻ may be takento obtain two new matrices Ψ_(j) ⁺ and Ψ_(j) ⁻ with complex elements. Itis to be noted thatΨ_(j)=Ψ_(j) ⁺+Ψ_(j) ⁻.

Then, a (2^(n)×2^(j+1))-complex matrix Φ_(j) may be created by fillingthe first half of the columns with the columns of Ψ_(j) ⁺ and the secondhalf of the columns with the columns of Ψ_(j) ⁻, but by also reversingthe order of the columns. For example, then Φ_(j) may be represented bythe matrix

${\Phi_{j} = \begin{bmatrix}{\Psi_{j}^{+}\left( {0,0} \right)} & \ldots & {\Psi_{j}^{+}\left( {0,{2^{j} - 1}} \right)} & {\Psi_{j}^{-}\left( {0,{2^{j} - 1}} \right)} & \ldots & {\Psi_{j}^{-}\left( {0,0} \right)} \\{\Psi_{j}^{+}\left( {1,0} \right)} & \ldots & {\Psi_{j}^{+}\left( {1,{2^{j} - 1}} \right)} & {\Psi_{j}^{-}\left( {1,{2^{j} - 1}} \right)} & \ldots & {\Psi_{j}^{-}\left( {1,0} \right)} \\\vdots & \; & \vdots & \vdots & \; & \vdots \\{\Psi_{j}^{+}\left( {{2^{n} - 2},0} \right)} & \ldots & {\Psi_{j}^{+}\left( {{2^{n} - 2},{2^{j} - 1}} \right)} & {\Psi_{j}^{-}\left( {{2^{n} - 2},{2^{j} - 1}} \right)} & \ldots & {\Psi_{j}^{-}\left( {{2^{n} - 2},0} \right)} \\{\Psi_{j}^{+}\left( {{2^{n} - 1},0} \right)} & \ldots & {\Psi_{j}^{+}\left( {{2^{n} - 1},{2^{j} - 1}} \right)} & {\Psi_{j}^{-}\left( {{2^{n} - 1},{2^{j} - 1}} \right)} & \ldots & {\Psi_{j}^{-}\left( {{2^{n} - 1},0} \right)}\end{bmatrix}},$where Ψ_(j) ^(±) (l, k) may be the (l,k)-th element of the matricesΨ_(j) ^(±).

Given a function fε

^(2n) representing N=2^(n) samples of a signal, the best approximationto f, in the least squares sense, for example, may then be obtainedusing the columns of the matrix Φ_(j). One way this can be achieved isby performing a singular value decomposition (“SVD”) of the matrix Φ_(j)and solving the system of the linear equationsΦ_(j) {right arrow over (c)}=f,where c may be the vector of coefficients and f may be viewed as acolumn vector.

As shown in FIG. 8, for example, two different compact cupolets (i.e.,as shown in parts (a) and (e)) may be used to approximate an arbitrarysignal f at different resolution-levels using a Walsh Transform-liketechnique according to the invention. For example, parts (b) and (f) ofFIG. 8, using the cupolets of parts (a) and (e), respectively, maycorrespond to the coarsest level (i.e., level 0) of approximation usingonly one basis element. Parts (c) and (g), on the other hand, maycorrespond to level 4 of approximation using 16 basis elements, forexample. It may be seen that the fit in part (c) is converging morerapidly than the fit in part (b), for example, and that the overallstructure of the signal f is fit reasonably in the coarsest level, butthat the finer structures of the signal are not fit until the higherlevels of approximation. For example, parts (d) and (h) may correspondto level 8 of approximation using 256 basis elements. In one embodiment,there may be 256 sample points in the signal f. Therefore, at thehighest resolution level (i.e., when the number of basis elements isequal to the dimension of the space), the fit may be exact (i.e., up tomachine precision).

When approximating a given signal f with a given compact cupolet c, thecomplexification process may introduce a complex number α, such thatf≈αc ⁺ + αc ⁻,where α may denote the complex conjugate of α. The term αc⁺+ αc⁻ can beregarded as a nonlinear phase deformation of the compact cupolet c thatcan result in the best approximation of the signal f with one complexparameter α.

The Walsh Transform-like technique can also be used in image compressionin accordance with the invention. For example, given an image, the imagemay first be transformed into a color space, such as YCbCr or YUV. Forthe ease of computations, it may be assumed that the image has sidelengths equal to powers of 2 in this example. Then, any suitablescanning method may be used to scan each layer of the image, including,but not limited to, horizontal scan, vertical scan, zig-zag scan (i.e.,on line one go left to right and then on line two go right to left andcontinue altering this pattern), any of the known scanning methods thatmay be applied to any of the sub-blocks of the image, and combinationsthereof, for example. In this embodiment, 3 one-dimensional signalsrepresenting each layer of the image may be obtained, for example. Usingsimilar techniques, as in windowed Fourier analysis, these signals maybe partitioned into desirable windows and each window may be representedusing the compact cupolet transform described above.

FIG. 9, for example, shows compressions of a sample image (i.e., parts(b), (c), and (d)) along with the original image (i.e., part (a)). Thesize of the original image used is 256×256 pixels and the size of thewindowed data is a single scan line or 256 points. The compressions ofparts (b), (c), and (d) of FIG. 9 are done in resolution-levels 4, 6,and 8, respectively, and the number of basis elements per data window ineach of these three compressions is 15, 63, and 254, respectively. Inthis example, the cupolet used to compress the image of part (a) of FIG.9 is the same as the cupolet in part (e) of FIG. 8. The spikes in thereconstructed signal in part (g) of FIG. 8 clearly explain the verticalline artifacts that appear in part (b) of FIG. 9.

There are several other possible ways of implementing this method forimage compression according to the invention, such as by using differentscanning techniques or by creating two-dimensional bases. For example, atwo-dimensional basis could be created by taking the “product” of acupolet in the horizontal direction, with a cupolet in the verticaldirection, such that the two-dimensional cupolets would represent asurface over the image plane. For example, if C1 is one cupolet and C2is another cupolet, then a two-dimensional version could be defined asD(x,y)=C1(x)·C2(y), such that for any pixel with indices (x,y), thevalue of D(x,y) could be the value (e.g., height) of the two-dimensionalcupolet at the point (x,y).

According to another embodiment of the invention, operating on a compactcupolet to produce bases for function representation may be accomplishedby defining a partial periodic multiresolution analysis (“PPMRA”) andconstructing a PPMRA using compact cupolets. For example, given acompact cupolet over a window of length N, this PPMRA technique may beaccomplished as follows.

First, the cupolet may be down-sampled to some number of points N/P,where P may be a factor of N. The number P may be small enough so thatthe cupolet does not lose its topological structure. This can provide ashorter vector that has the same shape as the original compact cupolet.Next, the cupolet may be padded with zeros such that it may have thesame length as the data window. This can simplify processing by allowingall vectors to have the same length. Then, the zero-padded cupolet maybe periodically extended to the entire real line. This zero-paddedcupolet may act as a scaling function that can be utilized to constructa multiresolution analysis. For example, the zero-padded cupolet may actas a scaling function that can be utilized in the same way that scalingfunctions are used to construct wavelet bases. A multiresolutionanalysis may then be used to approximate signals at differentresolutions. This process is now described in more detail.

In one embodiment of the invention, the construction of a PPMRA may beguided in part by “Petukhov, A.: ‘Periodic Discrete Wavelets,’ Algebra iAnaliz 8 (1996) 151-183,” which is hereby incorporated by referenceherein in its entirety. For example, the basis functions may be sampledN=2^(n) times in a data window, but they may be extended periodicallyoutside of the window. Therefore, in one embodiment, whenever an indexranges beyond N, the value may be taken from the periodic extension.

In accordance with an embodiment of the invention, a PPMRA may bedefined as follows. A sequence of linear function spaces {V^(j)}_(j=s)^(n), s≧0, may be referred to as a partial periodic multiresolutionanalysis (“PPMRA”) of the space

^(N), N=2^(n), if the following conditions or properties are satisfied:

-   -   (1) V^(s)⊂V^(s+1)⊂ . . . ⊂V^(j)⊂ . . . V^(n)=        ^(N); dim V^(j)=2^(j), j=s, . . . , n;    -   (2) If f(x)εV^(j), then f(2x)εV^(j+1);    -   (3) If f(x)εV^(j+1), then there exists g(x)εV^(j) such that

${{g\left( {2x} \right)} = {{f(x)} + {f\left( {x + \frac{N}{2}} \right)}}};$and

-   -   (4) The spaces V^(j), j=s, . . . , n, are invariant under the        shift by 2^(n−j) samples (i.e., for any function fεV^(j) and for        any kε        then f(x+k2^(n−j))εV^(j).        It should be noted that the index s may be bigger than 0 (i.e.,        the first resolution-level may include more than 1 basis        element). This is because the shifts at levels lower than s may        result in a degenerate cupolet representation. It should also be        noted that property (3) above may define the relationship        between a finer and coarser resolution. Therefore, this process        can be thought of as an up-sampling of the finer resolution, and        then truncation to N samples. For a given cupolet, and for a        fixed sample window length, a set of sampled versions may exist        that preserve the cupolet structure. This can define the total        number of available resolutions. For example, for N=2^(n) and a        cupolet that must have 2^(k) samples to preserve its structure,        the number P of non-degenerate resolutions may be given by the        following formula:

$P = {{1 + {\log_{2}\left( \frac{2^{n}}{2^{k}} \right)}} = {1 + n - {k.}}}$

For a function φ¹εV¹, l=s, . . . , φ¹ may be denoted by the vectorfunction{right arrow over (φ)}¹(x)=(φ¹(x),φ¹(x−2^(n−1)), . . .,φ¹(x−(2^(l)−1)2^(n−1)))^(T).In order to construct the basis for each resolution-level with a givencompact cupolet γ, the compact cupolet may first be rescaled andre-sampled. This may be done as follows according to one embodiment ofthe invention. For the given compact cupolet γ, assuming that it issampled N=2^(n) times over the data window, let φ^(s)=γ. The scalingfunction φ^(s) and its shift may define the coarsest resolution-level.The next finer resolution level may be defined by regenerating (i.e.,re-sampling) γ such that it has N/2 samples. Then, φ^(s+1)(m) may bedefined as follows:

${\varphi^{s + 1}(m)} = \left\{ \begin{matrix}{\gamma(m)} & {{{for}\mspace{14mu} 0} \leq m \leq {\frac{N}{2} - 1.}} \\0 & {{{for}\mspace{14mu}\frac{N}{2}} \leq k < {N.}}\end{matrix} \right.$This process may continue up to the finest available resolution. At thislevel, an integer n<n may be chosen, and the compact cupolet γ may beregenerated with 2 ^(n) points. Furthermore, the functionφ^(s+P)(m)=φ^(n)(m) may be defined by the following:

${\varphi^{s + P}(m)} = {{\varphi^{n}(m)} = \left\{ \begin{matrix}{\gamma(m)} & {{{{for}\mspace{14mu} 0} \leq m \leq {2^{\overset{\sim}{n}} - 1}},} \\0 & {{{for}\mspace{14mu} 2^{\overset{\sim}{n}}} \leq m < {2^{n}.}}\end{matrix} \right.}$

The functions φ^(i)(m), i=s, s+1, . . . , s+P may be normalized bydividing them with their Euclidean norm. FIG. 10, for example, shows afew of the functions φ^(n) and γ for the case n=10 and n=7. As shown,the functions in parts (a), (b), and (c) of FIG. 10 represent examplesof compact cupolets at the coarsest resolution, while the versions inparts (d), (e), and (f) of FIG. 10 correspond to the level 4 scalingfunctions of parts (a), (b), and (c), respectively. It should be notedthat the finest resolution-level may have 2^(n) basis elements.

In one embodiment of the invention, the scaling functions may satisfythe following recursion relation:φ^(j−1)(2m)=φ^(j)(m)+φ^(j)(m+2^(n−1)), j=s+1, . . . ,n.At each resolution-level j, a linearly independent set may beconstructed by shifting the corresponding scaling function by 2^(n−j)units. For example, by letting span (φ^(j)) denote the subspace V^(j)spanned by the components of φ^(j), this space can be used toapproximate all the functions fε

^(N) in a least squares sense. In other embodiments, rather than in aleast squares sense, a measure based on a psychoacoustic measure may beused for audio and a measure based on psychovisual measure may be goodfor video and images.

In one embodiment, let M^(j) be a matrix whose columns are shiftedversions of the scaling function φ^(i). For the j-th resolution-level,the scaling function φ^(i) may be shifted by m2^(n−j), for m=0, 1, . . ., 2^(j)−1. For example, let matrix M^(j) be defined as follows:M ^(j)=[φ^(j)(·),φ^(j)(·+2^(n−j)), . . . ,φ^(j)(·+(2^(j)−1)2^(n−j))],whereφ^(j)(·+m2^(n−j))=(φ^(j)(1+m2^(n−j)),φ^(j)(2+m2^(n−j)), . . .,φ^(j)(2^(n) +m2^(n−j)))^(T).Then, to get an approximation of function f in the space V^(j), thefollowing set of linear equations may be solved, for example, in theleast squares sense:M ^(j) X=f,where X=(x₁, . . . , x₂ ^(j)) are the coefficients. Therefore, f(x) maybe approximated as follows:

${f(x)} \approx {\sum\limits_{m = 1}^{2^{j}}{x_{m}{{\varphi^{j}\left( {x + {m\; 2^{n - j}}} \right)}.}}}$

However, in practice, the matrix M^(j) may not be of full rank whendoing these computations. Therefore, to overcome this problem, asingular value decomposition of the matrix M^(j) may be performedaccording to the invention. For example, it may be assumed that matrixM^(j)=UWV^(T), where U and V are orthogonal matrices. The degree ofsingularity of the matrix M^(j) can be determined by monitoring thediagonal elements of the matrix W that are close to zero (e.g., up tomachine precision). A sufficiently small positive number ε may be chosenand used to replace the zero diagonal elements. This can provide a newmatrix W, which in turn may give rise to a nonsingular matrix M ^(j).The new columns of the matrix M ^(j) may only be slightly perturbed fromthose in matrix M^(j).

Therefore, according to one embodiment of the invention, the new matrixM ^(j) may be used to solve the linear equation M ^(j)X=f, where thesolution for X can be interpreted as the spectrum of the compact cupolettransformation of the signal f. This spectrum of the signal f has verydesirable structure for compression purposes. FIG. 11, for example,shows an arbitrary signal f and its approximations at differentresolution-levels. In this embodiment, the actual length of the signalused is 256 samples and there are 4 different resolution-levelsavailable, each one shown in a respective part of the first column ofFIG. 11 (i.e., parts (a), (b), (c), and (d)). The number of basiselements in each of these resolution-levels is 32, 64, 128, and 256,respectively. The spectral values of a typical signal f may beoscillatory and bounded by an envelope function. This envelope functionmay be almost periodic and may only have low frequency components. Forexample, each one of parts (e), (f), (g), and (h) in the second columnof FIG. 11 shows the corresponding spectral values along with theenclosing envelope function for a respective one of the signalapproximations in the first column of FIG. 11 (i.e., parts (a), (b),(c), and (d)).

In accordance with an embodiment of the invention, the differentresolution-levels may be combined to incorporate the coarser elementswith the finer grained elements and achieve a more robust variation inapproximating a signal. For example, assuming that there are a totalnumber of 2^(k) basis elements, of length 2^(n), available in thecoarsest resolution-level, then there may be a total number of n−k+1available resolution-levels. A new basis matrix may be constructed bypicking certain basis elements from each of these resolution-levels. Inthis way, a matrix may be obtained that can fit the coarser structure aswell as the finer structures in a given signal.

To accomplish this, according to an embodiment of the invention, asubset of the columns from the M^(j) matrices may be selected at eachresolution-level. Since M^(j) has 2^(k) columns, 2^(l) columns may bechosen, where 1<k, and a matrix W¹ may be defined as follows:

$\begin{matrix}{W^{1} = \left\lbrack {{\varphi^{k}\left( {{\cdot {+ 0}} \times 2^{k - l}2^{n - k}} \right)},{\varphi^{k}\left( {{\cdot {+ 1}} \times 2^{k - l}2^{n - k}} \right)},\ldots\mspace{14mu},} \right.} \\\left. {\varphi^{k}\left( {{\cdot {+ \left( {2^{l} - 1} \right)}} \times 2^{k - l}2^{n - k}} \right)} \right\rbrack \\{= \left\lbrack {{\varphi^{k}\left( {{\cdot {+ 0}} \times 2^{n - l}} \right)},{\varphi^{k}\left( {{\cdot {+ 1}} \times 2^{n - l}} \right)},\ldots\mspace{14mu},} \right.} \\{\left. {\varphi^{k}\left( {{\cdot {+ \left( {2^{l} - 1} \right)}} \times 2^{n - l}} \right)} \right\rbrack.}\end{matrix}$Then, a sequence of matrices {W^(j)}_(j=2) ^(n−k+1) may be defined bypicking 2^(j+l−1) columns from the (k+j−1)-th basis matrix M^(k+j−1).For example, matrix W^(j) may be defined as follows:

$\begin{matrix}{W^{j} = \left\lbrack {{\varphi^{k + j - 1}\left( {{\cdot {+ 0}} \times 2^{k - l}2^{n - {({k + j - 1})}}} \right)},\ldots\mspace{14mu},} \right.} \\{{\varphi^{k + j - 1}\left( {{\cdot {+ 1}} \times 2^{k - l}2^{n - {({k + j - 1})}}} \right)},\ldots\mspace{11mu},} \\{\varphi^{k + j - 1}\left( {{\cdot {+ \left( {2^{j + l - 1} - 1} \right)}} \times 2^{k - l}2^{n - {({k + j - 1})}}} \right\rbrack} \\{= \left\lbrack {{\varphi^{k + j - 1}\left( {{\cdot {+ 0}} \times 2^{n - j - l + 1}} \right)},\ldots\mspace{14mu},} \right.} \\{{\varphi^{k + j - 1}\left( {{\cdot {+ 1}} \times 2^{n - j - l + 1}} \right)},\ldots\mspace{14mu},} \\{\varphi^{k + j - 1}\left( {{\cdot {+ \left( {2^{j + l - 1} - 1} \right)}} \times 2^{n - j - l + 1}} \right\rbrack}\end{matrix}$for j=2, . . . , n−k+1.

Next, these newly obtained matrices may be complexified using the sametechnique described above or in any other suitable way, for example. Theterm “complexify” may be referred to herein as any technique that maygive the system one complex (i.e., as in complex numbers using thesquare root of −1) degree of freedom, which may be the equivalent to tworeal degrees of freedom). This complexification can allow for some phaseadjustment of the different frequencies in the cupolet so that it mayneed not remain in its most-compact form. Although it is to be notedthat there are other suitable ways to allow the phase to vary inaccordance with the invention.

Then, a column of ones may be inserted at the beginning of the matrix.By performing a least squares approximation with this matrix, thecompact cupolets can go through a nonlinear phase deformation that mayadjust the compact cupolets to approximate the signal in the bestpossible way for this one-parameter phase deformation. It is to beunderstood, however, that while a least squares approximation may beused in certain embodiments, there are many other suitable types ofminimization functions that may be used in accordance with theinvention. In order to get higher resolution-levels, this same processcan be applied by picking more basis elements from each resolution-level(i.e., by choosing a smaller step size).

FIG. 12, for example, shows the approximation of an arbitrary signalusing a PPMRA technique according to the invention. The total number ofbasis elements used in the approximations of parts (a), (b), and (c) ofFIG. 12 are 4, 8, and 16, respectively, and the number of samples in thesignal is 256. As shown, the convergence can be extremely rapid for sucha small number of basis elements.

Similarly, this PPMRA technique can be used for compressing an image.FIG. 13, for example, shows the successive approximations of a sampleimage (i.e., parts (b), (c), and (d)) along with the original image(i.e., part (a)). The size of the original image used is 256×256 pixelsand the size of the windowed data is a single scan line or 256 points.The compressions of parts (b), (c), and (d) of FIG. 13 are done inresolution-levels 4, 6, and 8, respectively, and the number of basiselements used in each level is 24, 56, and 120 per data window,respectively.

When compared to previous images (e.g., the images of FIG. 9), thequality of the approximated image using 24 basis elements according tothis embodiment (e.g., part (b) of FIG. 13) may be considered roughlyequivalent to the quality of the previous image example using 63 basiselements according to the previous embodiment (e.g., part (c) of FIG.9). Furthermore, the quality of the approximated image using 56 basiselements according to this embodiment (e.g., part (c) of FIG. 13) may beconsidered essentially perfect. Therefore, by adding an extra degree offreedom associated with the phase deformation, the image reconstructionmay become of significantly higher quality using fewer basis elements.This effect may be particularly noticeable on full-sized images, forexample.

According to yet another embodiment of the invention, operating on acompact cupolet to produce bases for function representation may beaccomplished by conducting a periodic multiresolution analysis that maybe constructed with each resolution-level created by translations of anappropriately chosen scaling function. For example, given a compactcupolet, the cupolet may be down-sampled, as described above withrespect to another embodiment, for example. Then the down-sampledcupolet may be padded with zeros such that it may fit the data window.By shifting this compact cupolet by certain amounts through the datawindow and by using the shifted cupolet as basis elements, a chain oflinear subspaces may be realized that can be used to approximate signalsat different resolutions. For example, given a compact cupolet over awindow of length N, this scaling and shifting technique may beaccomplished as follows.

First, the cupolet may be down-sampled to some number of points N/P,where P may be a factor of N. The number P may be small enough so thatthe cupolet does not lose its topological structure. This can provide ashorter vector that has the same shape as the original compact cupolet.It is to be understood, that while P may be a factor of any number N, Pmay be a power of 2 in certain embodiments such that the scaling andshifting may work out evenly. For example, if the cupolet is scaled downby a factor of 2, and then the new version is shifted by half thewindow, the window may still be covered. However, other divisions may beused as well, such as by letting P=3 and then using shifts of ⅓, forexample, according to other embodiments of the invention.

Next, the cupolet may be padded with zeros such that it may fit in thedata window. This can simplify processing by allowing all vectors tohave the same length. Then, each zero-padded cupolet may be shifted bythe same factor (e.g., P) to generate a linear space. Each linear spacemay then be used as a resolution-level. Finally, a certain number ofbasis elements from each resolution-level may be chosen to generate amatrix of basis elements. This matrix may then be used to capture bothcoarser and finer structures of a given image. This process is nowdescribed in more detail.

In one embodiment of the invention, by letting c_(l) ¹ be a compactcupolet of length 2^(n), c_(l) ¹, may be viewed as the scaling functionfor the lowest (i.e., coarsest) resolution-level. That is, in oneembodiment, c_(l) ¹ may be representing the lowest frequencies in thespectrum of a given signal of length 2^(n). By letting V¹ be the closureof the linear space spanned by the vector c_(l) ¹, the higher (i.e.,finer) resolution-levels V^(j), j=2, . . . , 2^(n−1) may then be definedby introducing appropriate versions of the scaling function in eachresolution-level.

The scaling functions c_(j) ¹, j=2, . . . , 2^(n−1) may be definedrecursively according to an embodiment of the invention. For example, inorder to obtain the scaling functions c_(j) ¹, the original compactcupolet may be regenerated with a step size 2^(j−1) times bigger thanthe original step size. This may result in a down-sampling of thecompact cupolet c_(l) ¹. In one embodiment, if the windowed data is of alength 2^(n), the down-sampled compact cupolet may be shifted by 2^(n−j)points at a time, thereby producing 2^(j) orthogonal vectors c_(j) ¹,c_(j) ², . . . , c_(j) ² ^(j) . Similarly, V^(j) may be defined to bethe linear span of the orthogonal vectors c_(j) ¹, c_(j) ², . . . ,c_(j) ² ^(j) . This process may provide a total number of n availableresolution-levels. In order to approximate a given signal f, up to thek-th resolution-level, for example, a 2^(n)×(2^(k)−1) matrix M may beconstructed and defined as follows:M=[V ¹ ,V ² , . . . ,V ^(k)].Then, this matrix may be complexified (e.g., as described above withrespect to a previous technique), and a column of ones may be insertedat the beginning of the complexified matrix, for example.

FIG. 14, for example, shows the successive approximations of anarbitrary signal f at different resolution-levels using a scaling andshifting technique according to the invention. In one embodiment, thelength of the signal may be 256. The approximations of parts (a), (b),and (c) of FIG. 14 may be done up to resolution-levels 3, 5, and 7,respectively, and the number of basis elements in each of theseresolution levels is 7, 31, and 127, respectively.

Similarly, a scaling and shifting technique can be used for compressingan image in accordance with the invention. FIG. 15, for example, showsthe successive approximations of a sample image in differentresolution-levels (i.e., parts (b), (c), and (d)) along with theoriginal sample image (i.e., part (a)). The size of the original imageused is 256×256 pixels and the size of the windowed data is a singlescan line or 256 points. The approximations of parts (b), (c), and (d)of FIG. 15 are done in resolution-levels 4, 6, and 7, respectively, andthe number of basis elements used in each resolution-level is 15, 63,and 127 per data window, respectively.

While there have been described systems and methods for signal analysisusing orbits of a chaotic system, it is to be understood that manychanges may be made therein without departing from the spirit and scopeof the invention. For example, many other types of orbits besides thespecific types of cupolets described above may be used for signalanalysis according to the invention, such as any of the tens ofthousands of different cupolets that can be produced by any of thevarious types of chaotic signal generators. Moreover, because compactcupolets can have rich structure and a wide array of oscillatorybehaviors, it is to be understood that the most appropriate cupolet forthe desired analysis may be chosen by adding one or more pre-selectionsteps to any of the processes and techniques described above, such thatmore rapid convergence can be expected, for example. Those skilled inthe art will appreciate that the invention can be practiced by otherthan the described embodiments, which are presented for purposes ofillustration rather than of limitation, and the invention is limitedonly by the claims which follow.

1. A method for operating on a cupolet to produce bases for representing a signal, the method comprising: phase transforming the cupolet into a compact cupolet; extending the compact cupolet non-periodically over the entire real line by defining the compact cupolet to be zero outside a window of the compact cupolet; applying a transform to the extended compact cupolet to obtain at least a first basis, wherein the first basis is associated with a first set of one or more basis elements; and selecting at least one basis element from the first set to approximate the signal at a first resolution level.
 2. The method of claim 1 further comprising: applying a transform to the extended compact cupolet to obtain at least a second basis, wherein the second basis is associated with a second set of one or more basis elements; and selecting at least one basis element from the second set to approximate the signal at a second resolution level.
 3. The method of claim 1 further comprising: applying a transform to the extended compact cupolet to obtain at least a second basis, wherein the second basis is associated with a second set of one or more basis elements; and selecting at least two basis elements from the second set to approximate the signal at a second resolution level.
 4. The method of claim 1 further comprising: choosing the cupolet from a plurality of cupolets by comparing a spectrum of each of the plurality of cupolets with a spectrum of the signal.
 5. The method of claim 4, wherein the spectrum of each of the plurality of cupolets is a power spectrum.
 6. The method of claim 4, wherein the choosing comprises picking the cupolet with the spectrum closest to the spectrum of the signal in a least squares sense.
 7. The method of claim 1 further comprising: before applying the transform, non-linearly phase deforming the compact cupolet with at least one complex parameter.
 8. The method of claim 1, wherein the transform is a Walsh-like transform.
 9. The method of claim 1, wherein the signal is at least one of an image signal, an audio signal, and a video signal.
 10. A method for operating on a cupolet to produce bases for representing a signal, the method comprising: phase transforming the cupolet into a compact cupolet; down-sampling the compact cupolet by a factor of the length of a window of the compact cupolet; padding the down-sampled compact cupolet to have the same length as the length of the window of the compact cupolet; extending the padded cupolet periodically to the entire real line; sampling the extended cupolet to obtain at least a first basis, wherein the first basis is associated with a first set of one or more basis elements; and selecting at least one basis element from the first set to approximate the signal at a first resolution level.
 11. The method of claim 10 further comprising: sampling the extended cupolet to obtain at least a second basis, wherein the second basis is associated with a second set of one or more basis elements; and selecting at least one basis element from the second set to approximate the signal at a second resolution level.
 12. The method of claim 10 further comprising: sampling the extended cupolet to obtain at least a second basis, wherein the second basis is associated with a second set of one or more basis elements; and selecting at least two basis elements from the second set to approximate the signal at a second resolution level.
 13. The method of claim 10 further comprising: choosing the cupolet from a plurality of cupolets by comparing a spectrum of each of the plurality of cupolets with a spectrum of the signal.
 14. The method of claim 13, wherein the spectrum of each of the plurality of cupolets is a power spectrum.
 15. The method of claim 13, wherein the choosing comprises picking the cupolet with the spectrum closest to the spectrum of the signal in a least squares sense.
 16. The method of claim 10 further comprising: before applying the transform, non-linearly phase deforming the compact cupolet with at least one complex parameter.
 17. The method of claim 10, wherein the signal is at least one of an image signal, an audio signal, and a video signal.
 18. A method for operating on a cupolet to produce bases for representing a signal, the method comprising: phase transforming the cupolet into a compact cupolet; down-sampling the compact cupolet by a first factor of the length of a window of the compact cupolet to provide a first down-sampled cupolet; padding the first down-sampled cupolet to fit in the window of the compact cupolet; shifting the padded first down-sampled cupolet by the first factor to generate at least a first basis, wherein the first basis is associated with a first set of one or more basis elements; and selecting at least one basis element from the first set to approximate the signal at a first resolution level.
 19. The method of claim 18 further comprising: down-sampling the compact cupolet by a second factor of the length of the window of the compact cupolet to provide a second down-sampled cupolet, wherein the second factor is one of a factor of the first factor and a multiple of the first factor; padding the second down-sampled cupolet to fit in the window of the compact cupolet; shifting the padded second down-sampled cupolet by the second factor to generate at least a second basis, wherein the second basis is associated with a second set of one or more basis elements; and selecting at least one basis element from the second set to approximate the signal at a second resolution level.
 20. The method of claim 18 further comprising: down-sampling the compact cupolet by a second factor of the length of the window of the compact cupolet to provide a second down-sampled cupolet, wherein the second factor is one of a factor of the first factor and a multiple of the first factor; padding the second down-sampled cupolet to fit in the window of the compact cupolet; shifting the padded second down-sampled cupolet by the second factor to generate at least a second basis, wherein the second basis is associated with a second set of one or more basis elements; and selecting at least two basis elements from the second set to approximate the signal at a second resolution level.
 21. The method of claim 19 further comprising: generating a matrix using the at least one basis element from the first basis and the at least one basis element from the second basis; and using the matrix to approximate the signal at various resolution levels.
 22. The method of claim 18 further comprising: choosing the cupolet from a plurality of cupolets by comparing a spectrum of each of the plurality of cupolets with a spectrum of the signal.
 23. The method of claim 22, wherein the spectrum of each of the plurality of cupolets is a power spectrum.
 24. The method of claim 22, wherein the choosing comprises picking the cupolet with the spectrum closest to the spectrum of the signal in a least squares sense.
 25. The method of claim 18 further comprising: before applying the transform, non-linearly phase deforming the compact cupolet with at least one complex parameter.
 26. The method of claim 18, wherein the signal is at least one of an image signal, an audio signal, and a video signal.
 27. The method of claim 1, wherein the window of the compact cupolet is defined over a continuous interval of the real line.
 28. The method of claim 1, wherein phase transforming the cupolet into a compact cupolet comprises aligning at least a portion of the frequency components of the cupolet around a single global maximum.
 29. The method of claim 28, wherein phase transforming the cupolet into a compact cupolet further comprises phase transforming the compact cupolet into a less-compact cupolet with at least one complex parameter.
 30. The method of claim 29, wherein applying the transform comprises using the less-compact cupolet to obtain the at least a first basis.
 31. The method of claim 1, wherein phase transforming the cupolet into a compact cupolet comprises aligning all of the frequency components of the cupolet around a single global maximum. 